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Introduction to Constructive Type Theory CS6180: Introduction to Constructive Type Theory Robert Constable August 29, 2017 1 Course Summary and List of Topics These notes briefly summarize the content and course mechanics for fall 2017 CS6180. Constructive type theory is currently \hot," so this brief introduction highlights only a few reasons why type the- ory is important in contemporary computer science. In the course we explore several other reasons, and we will speculate on the future of the subject and its ties to related topics in programming languages, AI, theory, systems, and information science. 1.1 course requirements There will be lecture notes and readings covering nearly all of the material. Students can participate at various levels. Those who simply want to know type theory at a high level and hear about its potential and most interesting results can meet the requirements by writing a paper on the subject and submitting three of the suggested exercises. In addition all students can help by proof reading and making small extensions for one or two of the lecture notes among the 20 to 25 that will be posted. Participation at that level will see grades in the A to A- range. Students who are more interested can help in the exposition by writing tutorial notes on some of the conceptually most difficult material such as Brouwer's Bar Induction principle or his Fan Theorem or about the need for free choice sequences and their value in computer science. Students willing invest more and aspiring to an A+ grade and possibly a publishable article can explore one of the several research questions that will come up regularly. For example, type theory is not well developed for proving computational complexity results. On the other hand there have been suggestions for how to formalize complexity arguments which would make a good start and be the basis for a possibly publishable result (ppr). A fresh look at the issues would definitely merit a project paper. There are some new issues arising from the fact that the type theory we will study implements Brouwer's notion of free choice sequences. There should be opportunities for new results about these interesting constructive objects. Some questions along these lines will be proposed. 1 Other suggestions are mentioned later in these introductory notes. 1.2 proof assistants Constructive type theory is \implemented" in proof assistants. The course will briefly discuss and compare proof assistants. These implementations \bring type theories to life" and enable users to apply them to designing and coding algorithms and software systems and proving that they have certain properties. Knowing properties of algorithms and systems helps us understand their capabilities and limitations. We can compare those system properties to what we intend to do. As a very simple example, if we write an algorithm to recognize whether a given natural number is prime or composite, we expect to be able to prove based on the code that when the program is given a natural number, it will halt with a Boolean value telling us whether the input is prime or not. The code might produce factors if the number is not prime, depending on how we specify the task. Given that we have a very precise mathematical account of this task, we expect to be able to prove that the code does the right thing even if the argument is mathematically complex. 1.3 research questions One of the research themes we will investigate is how to use synthetic geometry, as in Euclid, to develop algorithms in computational geometry that are provably correct and efficient. A major effort worthy of summer support is to create a \compiler" from synthetic algorithms to analytical algorithms (using the constructive real numbers). Dr. Bickford has a working example of this approach that he will lecture about. A related topic would be to apply a result from homotopy theory to a practical geometric algorithm. The PRL group researcher Ariel Kellison is working on developing Nuprl material on Euclidean geometry to establish outreach to Ithaca High School. Many interesting research questions have come up in this project, and she and I could supervise a project in this area. In some cases we expect to know the cost of executing the algorithm. We will see why it is challenging to prove computational complexity bounds on algorithms in type theory. There have been PhD theses using Nuprl on this topic [20, 21, 13]. Since that work, we have considered other approaches worth exploring. They will come up during the course. The more complex an algorithm, the more we require ways to guarantee that it does the right thing. The importance of correctness is sometimes made by reporting on the consequences of trusting faulty algorithms, e.g. the Intel bug in a chip for arithmetic. There are books built around such stories [79, 90]. Type checking algorithms for programming languages with rich type systems help us understand programs. Model checking techniques systematically search for errors using a model of the task. Proof assistants work by helping designers create a formal proof that code has certain properties that can be precisely expressed in the prover's specification language. Typically these proofs provide much stronger guarantees than the type checkers do. We will also very briefly discuss previous strong (misguided) criticism of proof assistants, as in the 1979 article by De Millo, Lipton, and Perlis [42].1 1This article is reminiscent of the attacks on AI by the Dreyfus brothers which now seem so woefully misguided 2 Proof assistants provide considerable help with getting the many details of a formal proof correct. A good proof assistant can even extract algorithms that express the computational content of a proof. In that case, we end up with two results for the price of one. First, we find an algorithm that solves a problem. Second, we create a detailed formally checked explanation of why the algorithm is correct. One reason to implement a very expressive type theory is to provide the capability to precisely define a large variety of computing tasks. It is impossible that we will ever capture fully and completely the capabilities of a complex software system. There are limits that we know from G¨odel'sfamous incompleteness theorem [53]. On the other hand, rich expressiveness is a very worthy goal, and it motivates providing a formal system that can express as much computer science as possible. That is one sense in which constructive type theory is appropriate. It is the richest relevant formal theory we know. This lends credibility to the idea that constructive type theory could be an appropriate foundation for computer science; we'll see that it touches essentially all of the subjects in modern computer science. We will examine this theme throughout the course, starting now at the very beginning. Different versions of type theory are implemented in various proof assistants. We will mention three of them from time to time: Agda [25] from Sweden, Coq [22] from France, and Nuprl [37] from the US. Moreover, at least three new proof assistants are being built to implement variants of constructive type theory { JonPRL and RedPRL at CMU and Lean at Microsoft Research.2 There are also proof assistants for higher order logic (HOL). One of them is called HOL [55, 60, 70, 69, 50, 32]. Although this is not a course on proof assistants, we will illustrate their role with Nuprl examples from time to time. We already know that constructive type theory, like all sufficiently rich formal mathematical theories, cannot be complete in the sense that it is unable to prove all of its true statements. This is the consequence of G¨odel'sfamous incompleteness theorem [53]. For type theory, we need to be clear about what \intuitively true" means. We will consider this question during the course. 1.4 constructive type theories Why is type theory used in proof assistants? Could we use set theory instead, the foundational language of mathematics? The simplest and most naive answer is that we build software systems with programming languages, and they use types to specify tasks, not sets. Indeed, our work on type theory arose naturally from our efforts to design and implement a programming logic for the Cornell programming language PLC. Types were a key feature of this language, so we had to learn to formalize them. If our primary concern is the specification and correctness of algorithms and systems, type theory is the most appropriate choice. Some researchers see this as a mistake in that it complicates connections to a vast body of mathematics. Researchers with this concern in mind have shown how to express set theory in type theory. Peter Aczel has done very interesting work along these lines [4]. Furthermore, the practical value of expressing computational mathematics in [45]. The views of George B. Dyson [47] are much more prescient. One thing he wrote in his book Darwin among the Machines is this famous line: \In the game of life and evolution, there are three players at the table: human beings, nature, and machines. I am firmly on the side of nature. But nature, I suspect, is on the side of machines". 2Lean seems to have another name as well, but not with \PRL" in it! 3 programming languages has led to incorporating most mathematical concepts within type theory. We will see many examples of this during the course. For areas of mathematics that are concerned with constructions and algorithms, set theory was never \just right". Some constructive mathematicians started using notions such as \species" and others used the term \type" still others used \constructive sets" and so forth.
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